Invariant algebras and major indices for classical Weyl groups

Given a classical Weyl group W, that is, a Weyl group of type A, B or D, one can associate with it a polynomial with integral coefficients ZW given by the ratio of the Hilbert series of the invariant algebras of the natural action of W and Wt on the ring of polynomials C[x1, ...,xn]otimes t. We introduce and study several statistics on the classical Weyl groups of type B and D and show that they can be used to give an explicit formula for ZDn. More precisely, we define two Mahonian statistics, that is, statistics having the same distribution as the length function, Dmaj and ned on Dn. The statistic Dmaj, defined in a combinatorial way, has an analogous algebraic meaning to the major index for the symmetric group and the flag-major index of Adin and Roichman for Bn; namely, it allows us to find an explicit formula for ZDn. Our proof is based on the theory of t-partite partitions introduced by Gordon and further studied by Garsia and Gessel. Using similar ideas, we define the Mahonian statistic ned also on Bn and we find a new and simpler proof of the Adin-Roichman formula for ZBn. Finally, we define a new descent number Ddes on Dn so that the pair (Ddes,Dmaj) gives a generalization to Dn of the Carlitz identity on the Eulerian-Mahonian distribution of descent number and major index on the symmetric group.